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When you're dealing with standard normal distribution, the term "z-value" comes up quite a bit. A z-value helps you understand how far away a particular point is from the mean, using standard deviations as your unit of measurement. In a standard normal distribution, the mean is zero and the standard deviation is one.

As an example, a z-value of 1.71 means that you are 1.71 standard deviations above the mean. Similarly, a z-value of -1.71 tells you that you're 1.71 standard deviations below the mean. A higher absolute value of a z-value indicates that the point is further away from the mean.

Understanding z-values is crucial because they allow you to assess probabilities within a standard normal distribution, and are the backbone of standard deviation comparisons among different datasets.

Calculating probabilities in a standard normal distribution using z-values requires a grasp of some important steps.

Here’s how you can calculate probabilities between two z-values:

• First, find the probability that corresponds to each z-value using a standard normal table. This gives you the likelihood of a value being less than a given point under the normal distribution curve.
• For positive z-values, read directly off the table. For negative z-values, use the symmetry of the distribution. The probability for a negative z-value is the complement of the corresponding positive z-value. This is due to the properties of the bell curve, which is symmetric around the mean.
• Subtract the smaller probability from the larger to get the probability between the two z-values.

For instance, if the probability for a z-value of 1.71 is 0.9564, and the probability for -1.71 is found to be 0.0436, the probability of z being between these two points is 0.9564 - 0.0436. Through such calculations, one can account for the portion of data lying within a particular range in a standard normal distribution.

A standard normal table is a handy tool that makes finding probabilities easier when working with the normal distribution. This table contains a list of z-values and their corresponding cumulative probabilities – that is, the probability of a z-score or less occurring.

To use a standard normal table, follow these steps:

• Locate the row corresponding to the first decimal and integer part of the z-value. For instance, for a z-value of 1.71, you'd find the row for 1.7.
• Then, find the column corresponding to the hundredths place of the z-value. For 1.71, you’d seek the column labelled '.01'.
• The intersection of this row and column provides the cumulative probability.

Remember, it normally lists only positive z-values; for negative z-values, utilize the table’s symmetry property to find the probability. By understanding how a standard normal table functions, you can solve many problems concerning probabilities in a standard normal distribution efficiently.